Mathematics · Textbook solutions

Conic Sections

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 179 questions

7.1 Parabola

8 q

Solved Examples

Worked · 8
  1. Find the coordinates of the focus, equation of the directrix, length of latus rectum and coordinates of end points of latus rectum of each of the following parabolas.
    7.1.10 SolvedEx.1(i)
    y2=28xy^2 = 28x
  2. 7.1.10 SolvedEx.1(ii)
    3x2=8y3x^2 = 8y
  3. 7.1.10 SolvedEx.2
    Find the equation of the parabola with vertex at the origin, axis along Y-axis and passing through the point (6,−3)(6, -3)
  4. 7.1.10 SolvedEx.3
    Find the equation of the parabola whose diretrix is x+3=0x + 3 = 0
  5. 7.1.10 SolvedEx.4
    Calculate the focal distance of point P on the parabola y2=20xy^2 = 20x whose ordinate is 10
  6. 7.1.10 SolvedEx.5
    Find the equation of the parabola having (4,−8)(4, -8) as one of extremities of porabola.
  7. 7.1.10 SolvedEx.6
    For the parabola 3y2=16x3y^2 = 16x, find the parameter of the point (3,−4)(3, -4)
  8. 7.1.10 SolvedEx.7
    Find the coordinates of the vertex and focus, the equation of the axis of symmetry, diretrix and tangent at the vertex of the parabola x2+4x+4y+16=0x^2 + 4x + 4y + 16 = 0

7.1 Tangents to a Parabola

3 q

Solved Examples

Worked · 3
  1. 7.1.13 SolvedEx.1
    Find the equation of tangent to the parabola y2=9xy^2 = 9x at (1,−3)(1, -3).
  2. 7.1.13 SolvedEx.2
    Find the equation to tangent to the parabola y2=12xy^2 = 12x from the point (2,5)(2, 5).
  3. 7.1.13 SolvedEx.3
    Show that the tangents drawn from the point (−4,−9)(-4, -9) to the parabola y2=16xy^2 = 16x are perpendicular to each other.

Exercise 7.1

28 q
  1. Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola.
    Ex 7.1 Q1(i)
    5y2=24x5y^2 = 24x
  2. Ex 7.1 Q1(ii)
    y2=−20xy^2 = -20x
  3. Ex 7.1 Q1(iii)
    3x2=8y3x^2 = 8y
  4. Ex 7.1 Q1(iv)
    x2=−8yx^2 = -8y
  5. Ex 7.1 Q1(v)
    3y2=−16x3y^2 = -16x
  6. Ex 7.1 Q2
    Find the equation of the parabola with vertex at the origin, axis along Y-axis and passing through the point (−10,−5)(-10, -5).
  7. Ex 7.1 Q3
    Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (3,4)(3, 4).
  8. Ex 7.1 Q4
    Find the equation of the parabola whose vertex is O (0,0)(0, 0) and focus at (−7,0)(-7, 0).
  9. Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point.
    Ex 7.1 Q5(i)
    (1,−6)(1, -6)
  10. Ex 7.1 Q5(ii)
    (2,3)(2, 3)
  11. For the parabola 3y2=16x3y^2 = 16x, find the parameter of the point.
    Ex 7.1 Q6(i)
    (3,−4)(3, -4)
  12. Ex 7.1 Q6(ii)
    (27,−12)(27, -12)
  13. Ex 7.1 Q7
    Find the focal distance of a point on the parabola y2=16xy^2 = 16x whose ordinate is 2 times the abscissa.
  14. Find coordinate of the point on the parabola. Also find focal distance.
    Ex 7.1 Q8(i)
    y2=12xy^2 = 12x whose parameter is 13\frac{1}{3}
  15. Ex 7.1 Q8(ii)
    2y2=7x2y^2 = 7x whose parameter is −2-2
  16. Ex 7.1 Q9
    For the parabola y2=4xy^2 = 4x, find the coordinate of the point whose focal distance is 17.
  17. Ex 7.1 Q10
    Find length of latus rectum of the parabola y2=4axy^2 = 4ax passing through the point (2,−6)(2, -6).
  18. Ex 7.1 Q11
    Find the area of the triangle formed by the line joining the vertex of the parabola x2=12yx^2 = 12y to the end points of latus rectum.
  19. Ex 7.1 Q12
    If a parabolic reflector is 20 cm in diameter and 5 cm deep, find its focus.
  20. Ex 7.1 Q13
    Find coordinate of focus, vertex and equation of directrix and the axis of the parabola y=x2−2x+3y = x^2 - 2x + 3.
  21. Find the equation of tangent to the parabola.
    Ex 7.1 Q14(i)
    y2=12xy^2 = 12x from the point (2,5)(2, 5)
  22. Ex 7.1 Q14(ii)
    y2=36xy^2 = 36x from the point (2,9)(2, 9)
  23. Ex 7.1 Q15
    If the tangent drawn from the point (−6,9)(-6, 9) to the parabola y2=kxy^2 = kx are perpendicular to each other, find kk.
  24. Ex 7.1 Q16
    Two tangents to the parabola y2=8xy^2 = 8x meet the tangents at the vertex in the point P and Q. If PQ =4= 4, prove that the equation of the locus of the point of intersection of two tangent is y2=8(x+2)y^2 = 8(x + 2).
  25. Ex 7.1 Q17
    Find the equation of common tangent to the parabola y2=4xy^2 = 4x and x2=32yx^2 = 32y.
  26. Ex 7.1 Q18
    Find the equation of the locus of a point, the tangents from which to the parabola y2=18xy^2 = 18x are such that some of their slopes is −3-3.
  27. Ex 7.1 Q19
    The tower of a bridge, hung in the form of a parabola have their tops 30 meters above the road way and are 200 meters apart. If the cable is 5 meters above the road way at the centre of the bridge, find the length of the vertical supporting cable from the centre.
  28. Ex 7.1 Q20
    A circle whose centre is (4,−1)(4, -1) passes through the focus of the parabola x2+16y=0x^2 + 16y = 0. Show that the circle touches the directrix of the parabola.

7.2 Ellipse

6 q

Solved Examples

Worked · 6
  1. Find the coordinates of the foci, the vertices, the length of major axis, the eccentricity and the length of the latus rectum of the ellipse
    7.2.2 SolvedEx.1(i)
    x216+y29=1\frac{x^2}{16}+\frac{y^2}{9}=1
  2. 7.2.2 SolvedEx.1(ii)
    4x2+3y2=14x^2+3y^2=1
  3. 7.2.2 SolvedEx.1(iii)
    3x2+4y2=13x^2+4y^2=1
  4. 7.2.2 SolvedEx.1(iv)
    4x2+9y2−16x+54y+61=04x^2+9y^2-16x+54y+61=0
  5. 7.2.2 SolvedEx.2
    Find the equation of an ellipse having vertices (±13,0)(\pm 13,0) and foci (±5,0)(\pm 5,0)
  6. 7.2.2 SolvedEx.3
    Find the eccentricity of an ellipse whose length of the latus rectum is one third of its minor axis.

7.2 Tangents and the Auxiliary Circle

4 q

Solved Examples

Worked · 4
  1. Find the equation of tangent to the ellipse
    7.2.8 SolvedEx.1(i)
    x28+y26=1\frac{x^2}{8}+\frac{y^2}{6}=1 at the point (2,3)(2,\sqrt{3}).
  2. 7.2.8 SolvedEx.1(ii)
    x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1 at the point whose eccentric angle is π4\frac{\pi}{4}.
  3. 7.2.8 SolvedEx.2
    Show that the line 2x+3y=122x+3y=12 is tangent to the ellipse 4x2+9y2=724x^2+9y^2=72.
  4. 7.2.8 SolvedEx.3
    Find the equations of tangents to the ellipse 4x2+9y2=364x^2+9y^2=36 passing through the point (2,−2)(2,-2).

Exercise 7.2

35 q
  1. Find the (i) lengths of the principal axes. (ii) co-ordinates of the focii (iii) equations of directrics (iv) length of the latus rectum (v) distance between focii (vi) distance between directrices of the ellipse:
    Ex 7.2 Q1(a)
    x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1
  2. Ex 7.2 Q1(b)
    3x2+4y2=123x^2 + 4y^2 = 12
  3. Ex 7.2 Q1(c)
    2x2+6y2=62x^2 + 6y^2 = 6
  4. Ex 7.2 Q1(d)
    3x2+4y2=13x^2 + 4y^2 = 1
  5. Find the equation of the ellipse in standard form if
    Ex 7.2 Q2(i)
    eccentricity =38= \frac{3}{8} and distance between its focii =6= 6.
  6. Ex 7.2 Q2(ii)
    the length of major axis 10 and the distance between focii is 8.
  7. Ex 7.2 Q2(iii)
    distance between directrix is 18 and eccentricity is 13\frac{1}{3}.
  8. Ex 7.2 Q2(iv)
    minor axis is 16 and eccentricity is 13\frac{1}{3}.
  9. Ex 7.2 Q2(v)
    the distance between foci is 6 and the distance between directrix is 503\frac{50}{3}.
  10. Ex 7.2 Q2(vi)
    The latus rectum has length 6 and foci are (±2,0)(\pm 2, 0).
  11. Ex 7.2 Q2(vii)
    passing through the points (−3,1)(-3, 1) and (2,−2)(2, -2)
  12. Ex 7.2 Q2(viii)
    the dist. between its directrix is 10 and which passes through (−5,2)(-\sqrt{5}, 2)
  13. Ex 7.2 Q2(ix)
    eccentricity is 23\frac{2}{3} and passes through (2,−53)\left(2, -\frac{5}{3}\right).
  14. Ex 7.2 Q3
    Find the eccentricity of an ellipse, if the length of its latus rectum is one third of its minor axis.
  15. Ex 7.2 Q4
    Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its focii.
  16. Ex 7.2 Q5
    Show that the product of the lengths of ht perpendicular segments drawn from the foci to any tangent line to the ellipse x225+y216=1\frac{x^2}{25} + \frac{y^2}{16} = 1 is equal to 16.
  17. Ex 7.2 Q6
    A tangent having slop −12-\frac{1}{2} to the ellipse 3x2+4y2=123x^2 + 4y^2 = 12 intersects the X and Y axes in the points A and B respectively. If O is the origin, find the area of the triangle.
  18. Ex 7.2 Q7
    Show that the line x−y=5x - y = 5 is a tangent to the ellipse 9x2+16y2=1449x^2 + 16y^2 = 144. Find the point of contact.
  19. Ex 7.2 Q8
    Show that the line 8y+x=178y + x = 17 touches the ellipse x2+4y2=17x^2 + 4y^2 = 17. Find the point of contact.
  20. Ex 7.2 Q9
    Determine whether the line x+3y2=9x + 3y\sqrt{2} = 9 is a tangent to the ellipse x29+y24=1\frac{x^2}{9} + \frac{y^2}{4} = 1. If so, find the co-ordinates of the pt of contact.
  21. Ex 7.2 Q10
    Find k, if the line 3x+4y+k=03x + 4y + k = 0 touches 9x2+16y2=1449x^2 + 16y^2 = 144.
  22. Find the equation of the tangent to the ellipse
    Ex 7.2 Q11(i)
    x25+y24=1\frac{x^2}{5} + \frac{y^2}{4} = 1 passing through the point (2,−2)(2, -2).
  23. Ex 7.2 Q11(ii)
    4x2+7y2=284x^2 + 7y^2 = 28 from the pt (3,−2)(3, -2).
  24. Ex 7.2 Q11(iii)
    2x2+y2=62x^2 + y^2 = 6 from the point (2,1)(2, 1).
  25. Ex 7.2 Q11(iv)
    x2+4y2=9x^2 + 4y^2 = 9 which are parallel to the line 2x+3y−5=02x + 3y - 5 = 0.
  26. Ex 7.2 Q11(v)
    x225+y24=1\frac{x^2}{25} + \frac{y^2}{4} = 1 which are parallel to the line x+y+1=0x + y + 1 = 0.
  27. Ex 7.2 Q11(vi)
    5x2+9y2=455x^2 + 9y^2 = 45 which are ⊥\perp to the line 3x+2y+y=03x + 2y + y = 0.
  28. Ex 7.2 Q11(vii)
    x2+4y2=20x^2 + 4y^2 = 20, ⊥\perp to the line 4x+3y=74x + 3y = 7.
  29. Ex 7.2 Q12
    Find the equation of the locus of a point the tangents form which to the ellipse 3x2+5y2=153x^2 + 5y^2 = 15 are at right angles.
  30. Ex 7.2 Q13
    Tangents are drawn through a point P to the ellipse 4x2+5y2=204x^2 + 5y^2 = 20 having inclinations θ1\theta_1 and θ2\theta_2 such that tan⁡θ1+tan⁡θ2=2\tan\theta_1 + \tan\theta_2 = 2. Find the equation of the locus of P.
  31. Ex 7.2 Q14
    Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse.
  32. Ex 7.2 Q15
    P and Q are two points on the ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 with eccentric angles θ1\theta_1 and θ2\theta_2. Find the equation of the locus of the point of intersection of the tangents at P and Q if θ1+θ2=π2\theta_1 + \theta_2 = \frac{\pi}{2}.
  33. Ex 7.2 Q16
    The eccentric angles of two points P and Q the ellipse 4x2+y2=44x^2 + y^2 = 4 differ by 2π3\frac{2\pi}{3}. Show that the locus of the point of intersection of the tangents at P and Q is the ellipse 4x2+y2=164x^2 + y^2 = 16.
  34. Ex 7.2 Q17
    Find the equations of the tangents to the ellipse x216+y29=1\frac{x^2}{16} + \frac{y^2}{9} = 1, making equal intercepts on co-ordinate axes.
  35. Ex 7.2 Q18
    A tangent having slope −12-\frac{1}{2} to the ellipse 3x2+4y2=123x^2 + 4y^2 = 12 intersects the X and Y axes in the points A and B respectively. If O is the origin, find the area of the triangle.

7.3 Hyperbola

4 q

Solved Examples

Worked · 4
  1. Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola
    7.3.3 SolvedEx.1(i)
    x24−y212=1\frac{x^2}{4}-\frac{y^2}{12}=1
  2. 7.3.3 SolvedEx.1(ii)
    y29−x216=1\frac{y^2}{9}-\frac{x^2}{16}=1
  3. 7.3.3 SolvedEx.2
    Find the equation of the hyperbola with the centre at the origin, transverse axis 12 and one of the foci at (35,0)(3\sqrt{5},0)
  4. 7.3.3 SolvedEx.3
    Find the equation of the hyperbola referred to its principal axes whose distance between directrices is 185\frac{18}{5} and eccentricity is 53\frac{5}{3}.

7.3 Tangents and Asymptotes

4 q

Solved Examples

Worked · 4
  1. 7.3.9 SolvedEx.1
    Find the equation of the tangent to the hyperbola 2x2−3y2=52x^2-3y^2=5 at a point in the third quadrant whose abscissa is −2-2.
  2. 7.3.9 SolvedEx.2
    Show that the line 4x−3y=164x-3y=16 touches the hyperbola 16x2−25y2=40016x^2-25y^2=400. Find the co-ordinates of the point of contact.
  3. 7.3.9 SolvedEx.3
    If the line 2x+y+k=02x+y+k=0 is tangent to the hyperbola x26−y28=1\frac{x^2}{6}-\frac{y^2}{8}=1 then find the value of kk.
  4. 7.3.9 SolvedEx.4
    The line x−y+3=0x - y + 3 = 0 touches the hyperbola whose foci are (±41, 0)(\pm\sqrt{41},\ 0). Find the equation of the hyperbola.

Exercise 7.3

31 q
  1. Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola.
    Ex 7.3 Q1(i)
    x225−y216=1\frac{x^2}{25} - \frac{y^2}{16} = 1
  2. Ex 7.3 Q1(ii)
    x225−y216=−1\frac{x^2}{25} - \frac{y^2}{16} = -1
  3. Ex 7.3 Q1(iii)
    16x2−9y2=14416x^2 - 9y^2 = 144
  4. Ex 7.3 Q1(iv)
    21x2−4y2=8421x^2 - 4y^2 = 84
  5. Ex 7.3 Q1(v)
    3x2−y2=43x^2 - y^2 = 4
  6. Ex 7.3 Q1(vi)
    x2−y2=16x^2 - y^2 = 16
  7. Ex 7.3 Q1(vii)
    y225−x29=1\frac{y^2}{25} - \frac{x^2}{9} = 1
  8. Ex 7.3 Q1(viii)
    y225−x2144=1\frac{y^2}{25} - \frac{x^2}{144} = 1
  9. Ex 7.3 Q1(ix)
    x2100−y225=+1\frac{x^2}{100} - \frac{y^2}{25} = +1
  10. Ex 7.3 Q1(x)
    x=2sec⁡θ,y=23 tan⁡θx = 2\sec\theta,\quad y = 2\sqrt{3}\,\tan\theta
  11. Ex 7.3 Q2
    Find the equation of the hyperbola with centre at the origin, length of conjugate axis 10 and one of the foci (−7,0)(-7, 0).
  12. Ex 7.3 Q3
    Find the eccentricity of the hyperbola, which is conjugate to the hyperbola x2−3y2=3x^2 - 3y^2 = 3.
  13. Ex 7.3 Q4
    If ee and e′e' are the eccentricities of a hyperbola and its conjugate hyperbola respectively, prove that 1e2+1(e′)2=1\frac{1}{e^2} + \frac{1}{(e')^2} = 1
  14. Find the equation of the hyperbola referred to its principal axes.
    Ex 7.3 Q5(i)
    whose distance between foci is 10 and eccentricity 52\frac{5}{2}.
  15. Ex 7.3 Q5(ii)
    whose distance between foci is 10 and length of conjugate axis 6.
  16. Ex 7.3 Q5(iii)
    whose distance between directrices is 83\frac{8}{3} and eccentricity is 32\frac{3}{2}.
  17. Ex 7.3 Q5(iv)
    whose length of conjugate axis =12= 12 and passing through (1,−2)(1, -2).
  18. Ex 7.3 Q5(v)
    which passes through the points (6,9)(6, 9) and (3,0)(3, 0).
  19. Ex 7.3 Q5(vi)
    whose vertices are (±7,0)(\pm 7, 0) and end points of conjugate axis are (0,±3)(0, \pm 3).
  20. Ex 7.3 Q5(vii)
    whose foci are at (±2,0)(\pm 2, 0) and eccentricity 32\frac{3}{2}.
  21. Ex 7.3 Q5(viii)
    whose length of transverse and conjugate axis are 6 and 9 respectively.
  22. Ex 7.3 Q5(ix)
    whose length of transverse axis is 8 and distance between foci is 10.
  23. Find the equation of the tangent to the hyperbola.
    Ex 7.3 Q6(i)
    3x2−y2=43x^2 - y^2 = 4 at the point (2,22)(2, 2\sqrt{2}).
  24. Ex 7.3 Q6(ii)
    3x2−4y2=123x^2 - 4y^2 = 12 at the point (4,3)(4, 3).
  25. Ex 7.3 Q6(iii)
    x2144−y225=1\frac{x^2}{144} - \frac{y^2}{25} = 1 at the point whose eccentric angle is π3\frac{\pi}{3}.
  26. Ex 7.3 Q6(iv)
    x216−y29=1\frac{x^2}{16} - \frac{y^2}{9} = 1 at the point in a first quadratures whose ordinate is 3.
  27. Ex 7.3 Q6(v)
    9x2−16y2=1449x^2 - 16y^2 = 144 at the point L of latus rectum in the first quadrant.
  28. Ex 7.3 Q7
    Show that the line 3x−4y+10=03x - 4y + 10 = 0 is tangent till the hyperbola x2−4y2=20x^2 - 4y^2 = 20. Also find the point of contact.
  29. Ex 7.3 Q8
    If the 3x−4y=k3x - 4y = k touches the hyperbola x25−4y25=1\frac{x^2}{5} - \frac{4y^2}{5} = 1 then find the value of kk.
  30. Ex 7.3 Q9
    Find the equations of the tangents to the hyperbola x225−y29=1\frac{x^2}{25} - \frac{y^2}{9} = 1 making equal intercepts on the co-ordinate axes.
  31. Ex 7.3 Q10
    Find the equations of the tangents to the hyperbola 5x2−4y2=205x^2 - 4y^2 = 20 which are parallel to the line 3x+2y+12=03x + 2y + 12 = 0.

Miscellaneous Exercise 7

56 q

(I) Select the correct option

Practice · 20
  1. Misc I Q1
    The line y=mx+1y = mx + 1 is tangent to the parabola y2=4xy^2 = 4x if mm is ………\ldots\ldots\ldots
    1. A.
      1
    2. B.
      2
    3. C.
      3
    4. D.
      4
  2. Misc I Q2
    The length of latus rectum of the parabola x2−4x−8y+12=0x^2 - 4x - 8y + 12 = 0 is ………\ldots\ldots\ldots
    1. A.
      4
    2. B.
      6
    3. C.
      8
    4. D.
      10
  3. Misc I Q3
    If the focus of the parabola is (0,−3)(0, -3) its directrix is y=3y = 3 then its equation is
    1. A.
      x2=−12yx^2 = -12y
    2. B.
      x2=12yx^2 = 12y
    3. C.
      y2=12xy^2 = 12x
    4. D.
      y2=−12xy^2 = -12x
  4. Misc I Q4
    The coordinates of a point on the parabola y2=8xy^2 = 8x whose focal distance is 4 are ………\ldots\ldots\ldots
    1. A.
      (1/2, ±2)(1/2,\ \pm 2)
    2. B.
      (1, ±22)(1,\ \pm 2\sqrt{2})
    3. C.
      (2, ±4)(2,\ \pm 4)
    4. D.
      none of these
  5. Misc I Q5
    The end points of latus rectum of the parabola y2=24xy^2 = 24x are ………\ldots\ldots\ldots
    1. A.
      (6, ±12)(6,\ \pm 12)
    2. B.
      (12, ±6)(12,\ \pm 6)
    3. C.
      (6, ±6)(6,\ \pm 6)
    4. D.
      none of these
  6. Misc I Q6
    Equation of the parabola with vertex at the origin and diretrix x+8=0x + 8 = 0 is ………\ldots\ldots\ldots
    1. A.
      y2=8xy^2 = 8x
    2. B.
      y2=32xy^2 = 32x
    3. C.
      y2=16xy^2 = 16x
    4. D.
      x2=32yx^2 = 32y
  7. Misc I Q7
    The area of the triangle formed by the line joining the vertex of the parabola x2=12yx^2 = 12y to the end points of its latus rectum is ………\ldots\ldots\ldots
    1. A.
      22 sq.units
    2. B.
      20 sq.units
    3. C.
      18 sq.units
    4. D.
      14 sq.units
  8. Misc I Q8
    If P(π4)P\left(\frac{\pi}{4}\right) is any point on he ellipse 9x2+25y2=2259x^2 + 25y^2 = 225. S and S1S^1 are its foci then SP.S1P=S P . S^1 P =
    1. A.
      13
    2. B.
      14
    3. C.
      17
    4. D.
      19
  9. Misc I Q9
    The equation of the parabola having (2,4)(2, 4) and (2,−4)(2, -4) as end points of its latus rectum is ………\ldots\ldots\ldots
    1. A.
      y2=4xy^2 = 4x
    2. B.
      y2=8xy^2 = 8x
    3. C.
      y2=−16xy^2 = -16x
    4. D.
      x2=8yx^2 = 8y
  10. Misc I Q10
    If the parabola y2=4axy^2 = 4ax passes through (3,2)(3, 2) then the length of its latus rectum is ………\ldots\ldots\ldots
    1. A.
      23\frac{2}{3}
    2. B.
      43\frac{4}{3}
    3. C.
      13\frac{1}{3}
    4. D.
      4
  11. Misc I Q11
    The eccentricity of rectangular hyperbola is
    1. A.
      12\tfrac{1}{2}
    2. B.
      1/(2 12)1 / (2\ \tfrac{1}{2})
    3. C.
      2 122\ \tfrac{1}{2}
    4. D.
      1/(3 12)1 / (3\ \tfrac{1}{2})
  12. Misc I Q12
    The equation of the ellipse having foci (+4,0)(+ 4, 0) and eccentricity 13\frac{1}{3} is,
    1. A.
      9x2+16y2=1449x^2 + 16y^2 = 144
    2. B.
      144x2+9y2=1296144x^2 + 9y^2 = 1296
    3. C.
      128x2+144y2=18432128x^2 + 144y^2 = 18432
    4. D.
      144x2+128y2=18432144x^2 + 128y^2 = 18432
  13. Misc I Q13
    The equation of the ellipse having eccentricity 32\frac{\sqrt{3}}{2} and passing through (−8,3)(-8, 3) is
    1. A.
      4x2+y2=44x^2 + y^2 = 4
    2. B.
      x2+4y2=100x^2 + 4y^2 = 100
    3. C.
      4x2+y2=1004x^2 + y^2 = 100
    4. D.
      x2+4y2=4x^2 + 4y^2 = 4
  14. Misc I Q14
    If the line 4x−3y+k=04x - 3y + k = 0 touches the ellipse 5x2+9y2=455x^2 + 9y^2 = 45 then the value of k is
    1. A.
      + 21+\ 21
    2. B.
      ±321\pm 3\sqrt{21}
    3. C.
      + 3+\ 3
    4. D.
      + 3 (21)+\ 3\ (21)
  15. Misc I Q15
    The equation of the ellipse is 16x2+25y2=40016x^2 + 25y^2 = 400. The equations of the tangents making an angle of 180∘180^\circ with the major axis are
    1. A.
      x=4x = 4
    2. B.
      y=±4y = \pm 4
    3. C.
      x=−4x = -4
    4. D.
      x=±5x = \pm 5
  16. Misc I Q16
    The equation of the tangent to the ellipse 4x2+9y2=364x^2 + 9y^2 = 36 which is perpendicular to the 3x+4y=173x + 4y = 17 is,
    1. A.
      y=4x+6y = 4x + 6
    2. B.
      3y+4x=63y + 4x = 6
    3. C.
      3y=4x+653y = 4x + 6\sqrt{5}
    4. D.
      3y=x+253y = x + 25
  17. Misc I Q17
    Eccentricity of the hyperbola 16x2−3y2−32x−12y−44=016x^2 - 3y^2 - 32x - 12y - 44 = 0 is
    1. A.
      173\sqrt{\frac{17}{3}}
    2. B.
      193\sqrt{\frac{19}{3}}
    3. C.
      193\frac{\sqrt{19}}{3}
    4. D.
      173\frac{\sqrt{17}}{3}
  18. Misc I Q18
    Centre of the ellipse 9x2+5y2−36x−50y−164=09x^2 + 5y^2 - 36x - 50y - 164 = 0 is at
    1. A.
      (2,5)(2, 5)
    2. B.
      (1,−2)(1, -2)
    3. C.
      (−2,1)(-2, 1)
    4. D.
      (0,0)(0, 0)
  19. Misc I Q19
    If the line 2x−y=42x - y = 4 touches the hyperbola 4x2−3y2=244x^2 - 3y^2 = 24, the point of contact is
    1. A.
      (1,2)(1, 2)
    2. B.
      (2,3)(2, 3)
    3. C.
      (3,2)(3, 2)
    4. D.
      (−2,−3)(-2, -3)
  20. Misc I Q20
    The foci of hyperbola 4x2−9y2−36=04x^2 - 9y^2 - 36 = 0 are
    1. A.
      (±13, 0)(\pm\sqrt{13},\ 0)
    2. B.
      (±11, 0)(\pm\sqrt{11},\ 0)
    3. C.
      (±12, 0)(\pm\sqrt{12},\ 0)
    4. D.
      (0, ±12)(0,\ \pm\sqrt{12})

(II) Answer the following

Practice · 36
  1. For each of the following parabolas, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum.
    Misc II Q1(i)
    2y2=17x2y^2 = 17x
  2. Misc II Q1(ii)
    5x2=24y5x^2 = 24y
  3. Find the Cartesian co-ordinates of the points on the parabola y2=12xy^2 = 12x whose parameters are
    Misc II Q2(i)
    2
  4. Misc II Q2(ii)
    −3-3
  5. Misc II Q3
    Find the co-ordinates of a point of the parabola y2=8xy^2 = 8x having focal distance 10.
  6. Misc II Q4
    Find the equation of the tangent to the parabola y2=9xy^2 = 9x at the point (4,−6)(4, -6) on it.
  7. Misc II Q5
    Find the equation of the tangent to the parabola y2=8xy^2 = 8x at t=1t = 1 on it.
  8. Misc II Q6
    Find the equations of the tangents to the parabola y2=9xy^2 = 9x through the point (4,10)(4, 10).
  9. Misc II Q7
    Show that the two tangents drawn to the parabola y2=24xy^2 = 24x from the point (−6,9)(-6, 9) are at the right angle.
  10. Misc II Q8
    Find the equation of the tangent to the parabola y2=8xy^2 = 8x which is parallel to the line 2x+2y+5=02x + 2y + 5 = 0. Find its point of contact.
  11. Misc II Q9
    A line touches the circle x2+y2=2x^2 + y^2 = 2 and the parabola y2=8xy^2 = 8x. Show that its equation is y=±(x+2)y = \pm (x + 2).
  12. Misc II Q10
    Two tangents to the parabola y2=8xy^2 = 8x meet the tangent at the vertex in P and Q. If PQ=4PQ = 4, prove that the locus of the point of intersection of the two tangents is y2=8(x+2)y^2 = 8(x + 2).
  13. The slopes of the tangents drawn from P to the parabola y2=4axy^2 = 4ax are m1m_1 and m2m_2, show that, where k is a constant.
    Misc II Q11(i)
    m1−m2=km_1 - m_2 = k
  14. Misc II Q11(ii)
    (m1/m2)=k(m_1 / m_2) = k
  15. Misc II Q12
    The tangent at point P on the parabola y2=4axy^2 = 4ax meets the y−y-axis in Q. If S is the focus, show that SP subtends a right angle at Q.
  16. Find the (i) lengths of the principal axes (ii) co-ordinates of the foci (iii) equations of directrices (iv) length of the latus rectum (v) Distance between foci (vi) distance between directrices of the curve
    Misc II Q13(a)
    x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1
  17. Misc II Q13(b)
    16x2+25y2=40016x^2 + 25y^2 = 400
  18. Misc II Q13(c)
    x2144−y225=1\frac{x^2}{144} - \frac{y^2}{25} = 1
  19. Misc II Q13(d)
    x2−y2=16x^2 - y^2 = 16
  20. Find the equation of the ellipse in standard form if
    Misc II Q14(i)
    eccentricity =3/8= 3/8 and distance between its foci =6= 6.
  21. Misc II Q14(ii)
    the length of major axis 10 and the distance between foci is 8.
  22. Misc II Q14(iii)
    passing through the points (−3,1)(-3, 1) and (2,−2)(2, -2).
  23. Misc II Q15
    Find the eccentricity of an ellipse if the distance between its directrices is three times the distance between its foci.
  24. Misc II Q16
    For the hyperbola x2100−y225=1\frac{x^2}{100} - \frac{y^2}{25} = 1, prove that SA.S′A=25SA . S'A = 25, where S and S′S' are the foci and A is the vertex.
  25. Misc II Q17
    Find the equation of the tangent to the ellipse x25−y24=1\frac{x^2}{5} - \frac{y^2}{4} = 1 passing through the point (2,−2)(2, -2).
  26. Misc II Q18
    Find the equation of the tangent to the ellipse x2+4y2=100x^2 + 4y^2 = 100 at (8,3)(8, 3).
  27. Misc II Q21
    Show that the product of the lengths of its perpendicular segments drawn from the foci to any tangent line to the ellipse x225+y216=1\frac{x^2}{25} + \frac{y^2}{16} = 1 is equal to 16.
  28. Find the equation of the hyperbola in the standard form if
    Misc II Q22(i)
    Length of conjugate axis is 5 and distance between foci is 13.
  29. Misc II Q22(ii)
    eccentricity is 3/23/2 and distance between foci is 12.
  30. Misc II Q22(iii)
    length of the conjugate axis is 3 and distance between the foci is 5.
  31. Find the equation of the tangent to the hyperbola,
    Misc II Q23(i)
    7x2−3y2=517x^2 - 3y^2 = 51 at (−3,−2)(-3, -2)
  32. Misc II Q23(ii)
    x=3sec⁡θ, y=5tan⁡θx = 3\sec\theta,\ y = 5\tan\theta at θ=π/3\theta = \pi/3
  33. Misc II Q23(iii)
    x225−y216=1\frac{x^2}{25} - \frac{y^2}{16} = 1 at P(30∘)P(30^\circ).
  34. Misc II Q24
    Show that the line 2x−y=42x - y = 4 touches the hyperbola 4x2−3y2=244x^2 - 3y^2 = 24. Find the point of contact.
  35. Misc II Q25
    Find the equations of the tangents to the hyperbola 3x2−y2=483x^2 - y^2 = 48 which are perpendicular to the line x+2y−7=0x + 2y - 7 = 0
  36. Misc II Q26
    Two tangents to the hyperbola x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 make angles θ1\theta_1, θ2\theta_2, with the transverse axis. Find the locus of their point of intersection if tan⁡θ1+tan⁡θ2=k\tan\theta_1 + \tan\theta_2 = k.